Suppose $X=\mathrm{Spec}R$ where $R$ is a noetherian ring. We have the Serre functor $\mathcal{A}(M)$ of a module giving a coherent sheaf. Hartshorne spends a page proving that $\mathcal{A}(I)$ is a flasque sheaf for an injective module $I$. Isn't it true that $\mathcal{A}(I)$ is an *injective* object in the category of coherent sheaves?

In the end,he uses this to prove that the cohomology of a quasicoherent sheaf vanishes on a noetherian affine scheme. Wouldn't it have been easier simply to assert that $\mathcal{A}(I)$ is injective?

`$H^1(X,\mathcal{O}_X^\ast)$`

, it is important to work in the larger category. In any of these categories, flasque sheaves are acyclic for sheaf cohomology. – Jason Starr Dec 17 '12 at 18:04